Quantum geometric tensors from sub-bundle geometry

TL;DR

The paper addresses how to formulate quantum geometry when parameter spaces are curved and quantum states are constrained to a sub-bundle. It introduces a generalized quantum geometric tensor that includes an additional curvature contribution arising from a non-flat connection, and derives generalized Gauss–Codazzi–Mainardi equations for the sub-bundle geometry. The authors illustrate the framework with semiclassical Dirac fields in curved spacetime, showing how the extra curvature term enriches the quantum metric and its relation to Berry curvature. They further analyze Dirac fermions on the hyperbolic plane to demonstrate curvature-induced modifications of the quantum geometry and discuss implications for topological phases on curved lattices and in high-energy contexts.

Abstract

The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.